Unified analytical approach to the Darcy mixed convection with viscous dissipation in a vertical channel

The present paper is concerned with mixed convection in a vertical plane parallel channel filled with a porous medium. Steady parallel flow is examined, assuming that the effect of viscous dissipation is significant. Under these assumptions, the governing balance equations admit a first integral, so that the general solution can be given in an exact analytical form in terms of the Weierstrass' elliptic P-function. Based on this general solution and on a suitable parametrization of the problem, a unified approach which applies to all the thermal boundary conditions compatible with the steady parallel flow regime is reported. It is shown that the velocity field can either be unidirectional or bidirectional. Moreover, bidirectional flow configurations are possible also for vanishing average velocity, Um=0. A remarkable feature of the problem is that for Um smaller than Um,max, even two solution branches (dual solutions) exist, which merge when Um approaches its maximum value Um,max. The general features of the solution space, as well as the mechanical and thermal characteristics of the flow are discussed for two cases of physical and engineering interest (isoflux÷variable temperature, and isoflux÷isoflux wall conditions) in some detail.